Properties and applications of the distance functions on open sets

F. G. Avkhadiev

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2020), pp. 87-92.

Résumé

For an open subset of the Euclidean space of dimension $n$ we consider interior and exterior approximations by sequences of open sets. We prove convergence everywhere of the corresponding sequences of distance functions from boundary as well as convergence almost everywhere for their gradients. As applications we obtain several new Hardy-type inequalities that contain the scalar product of gradients of test functions and the gradient of the distance function from the boundary of an open subset of the Euclidean space.

Détail

Keywords: distance function, Rademacher theorem, Motzkin theorem, approximation of open set, Hardy type inequality.
Mots-clés : convex domain

Bibliographie

[1] Balinsky A.A., Evans W.D., Lewis R.T., The Analysis and Geometry of Hardy's Inequality, Universitext, Springer, Heidelberg–New York–Dordrecht–London, 2015 | DOI | MR | Zbl

[2] Rademacher H., “Über partielle und totale Differenzierbarkeit I”, Math. Ann., 89:4 (1919), 340–359 | DOI

[3] Federer H., Geometric function theory, Springer Verlag, New York, 1969 | MR

[4] Marcus M., Mitzel V.J., Pinchover Y., “On the best constant for Hardy's inequality in $\mathbb{R}^n$”, Trans. Amer. Math. Soc., 350 (1998), 3237–3250 | DOI | MR

[5] Avkhadiev F.G., “Hardy type inequalities in higher dimensions with explicit estimate of constants”, Lobachevskii J. Math., 21 (2006), 3–31 | MR | Zbl

[6] Avkhadiev F.G., Shafigullin I.K., “Tochnye otsenki konstant Khardi dlya oblastei so spetsialnymi granichnymi svoistvami”, Izv. vuzov. Matem., 2014, no. 2, 69–73 | Zbl

[7] Avkhadiev F.G., Wirths K.-J., “Unified Poincaré and Hardy inequalities with sharp constants for convex domains”, Z. Angew. Math. Mech. (ZAMM), 87:8-9 (2007, 632–642 | DOI | MR | Zbl

[8] Avkhadiev F.G., Wirths K.-J., “Weighted Hardy inequalities with sharp constants”, Lobachevskii J. Math., 31 (2010), 1–7 | DOI | MR | Zbl

[9] Avkhadiev F.G., Wirths K.-J., “Sharp Hardy-type inequalities with Lamb's constants”, Bull. Belg. Math. Soc. Simon Stevin, 18 (2011), 723–736 | DOI | MR | Zbl

[10] Avkhadiev F.G., “Konformno invariantnye neravenstva v oblastyakh evklidova prostranstva”, Izv. RAN. Ser. matem., 83:5 (2019), 3–26 | DOI | MR | Zbl

[11] Avkhadiev F.G., “Neravenstva tipa Khardi v ploskikh i prostranstvennykh otkrytykh mnozhestvakh”, Tr. matem. in-ta. im. V.A. Steklova, 255, 2006, 8–18 | Zbl

[12] Avkhadiev F.G., Nasibullin R.G., Shafigullin I.K., “Neravenstva tipa Khardi so stepennymi i logarifmicheskimi vesami v oblastyakh evklidova prostranstva”, Izv. vuzov. Matem., 2011, no. 9, 90–94 | Zbl

[13] Avkhadiev F.G., Nasibullin R.G., “Neravenstva tipa Khardi v proizvolnykh oblastyakh s konechnym vnutrennim radiusom”, Sib. matem. zhurn., 55:2 (2014), 191–200 | MR | Zbl

[14] Avkhadiev F.G., “Geometricheskoe opisanie oblastei, dlya kotorykh konstanta Khardi ravna $1/4$”, Izv. RAN. Ser. matem., 78:5 (2014), 3–26 | DOI | MR | Zbl

[15] Avkhadiev F.G., “Integralnye neravenstva v oblastyakh giperbolicheskogo tipa i ikh prilozheniya”, Matem. sb., 206:12 (2015), 3–28 | DOI | MR | Zbl

[16] Avkhadiev F.G., “Integralnye neravenstva Khardi i Rellikha v oblastyakh, udovletvoryayuschikh usloviyu vneshnei sfery”, Algebra i analiz, 30:2 (2018), 18–44

[17] Motzkin T.S., “Sur quelques propriétés charactéristiques des ensembles convexes”, Atti Real. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. Serie VI, 21 (1935), 562–567

[18] Mantegazza C., Mennucci A.C., “Hamilton-Jacobi equations and distance functions on Riemannian manifolds”, Appl. Math. Optim., 47 (2003), 1–25 | DOI | MR